The movement demanding that public interest institutions divest their holdings from fossil fuels is on a serious roll. Chapters have opened up in more than 100 US cities and states as well as on more than 300 campuses, where students are holding protests, debates and sit-ins to pressure their to rid their endowments of oil, gas and coal holdings. And under the "Fossil Free UK" banner, the movement is now crossing the Atlantic, with a major push planned by People & Planet for this summer. Some schools, including University College London, have decided not to wait and already have active divestment campaigns.

Though officially launched just six months ago, the movement can already claim some provisional victories: four US colleges have announced their intention to divest their endowments from fossil fuel stocks and bonds and, in late April, 10 US cities made similar commitments, including San Francisco (Seattle came on board months ago).

There are still all kinds of details to work out to toughen up these pledges, but the speed with which this idea has spread makes it clear that there was some serious pent-up demand. To quote the mission statement of the Fossil Free movement: "If it is wrong to wreck the climate, then it is wrong to profit from that wreckage. We believe that educational and religious institutions, city and state governments, and other institutions that serve the public good should divest from fossil fuels." I am proud to have been part of the group at 350.org that worked with students and other partners to develop the Fossil Free campaign. But I now realise that an important target is missing from the list: the environmental organisations themselves.

You can understand the oversight. Green groups raise mountains of cash every year on the promise that the funds will be spent on work that is attempting to prevent catastrophic global warming. Fossil fuel companies, on the other hand, are doing everything in their power to make the catastrophic inevitable. According to the UK's Carbon Tracker Initiative (on whose impeccable research the divestment movement is based), the fossil fuel sector holds five times more carbon in its reserves than can be burned while still leaving us a good shot of limiting warming to 2C. One would assume that green groups would want to make absolutely sure that the money they have raised in the name of saving the planet is not being invested in the companies whose business model requires cooking said planet, and which have been sabotaging all attempts at serious climate action for more than two decades. But in some cases at least, that was a false assumption.

Maybe that shouldn't come as a complete surprise, since some of the most powerful and wealthiest environmental organisations have long behaved as if they had a stake in the oil and gas industry. They led the climate movement down various dead ends: carbon trading, carbon offsets, natural gas as a "bridge fuel" – what these policies all held in common is that they created the illusion of progress while allowing the fossil fuel companies to keep mining, drilling and fracking with abandon. We always knew that the groups pushing hardest for these false solutions took donations from, and formed corporate partnerships with, the big emitters. But this was explained away as an attempt at constructive engagement – using the power of the market to fix market failures.

Now it turns out that some of these groups are literally part-owners of the industry causing the crisis they are purportedly trying to solve. And the money the green groups have to play with is serious. The Nature Conservancy, for instance, has $1.4bn (£900m) in publicly traded securities, and boasts that its piggybank is "among the 100 largest endowments in the country". The Wildlife Conservation Society has a $377m endowment, while the endowment of the World Wildlife Fund–US is worth $195m.

Let me be absolutely clear: plenty of green groups have managed to avoid this mess. Greenpeace, 350.org, Friends of the Earth, Rainforest Action Network, and a host of smaller organisations such as Oil Change International and the Climate Reality Project don't have endowments and don't invest in the stock market. They also either don't take corporate donations or place such onerous restrictions on them that extractive industries are easily ruled out. Some of these groups own a few fossil fuel stocks, but only so that they can make trouble at shareholder meetings.

The Natural Resources Defense Council is halfway there. It has a $118m endowment and, according to its accounting team, for direct investments "we specifically screen out extractive industries, fossil fuels, and other areas of the energy sector". However, the NRDC continues to hold stocks in mutual funds and other mixed assets that do not screen for fossil fuels. (The Fossil Free campaign is calling on institutions to "divest from direct ownership and any commingled funds that include fossil fuel public equities and corporate bonds within 5 years".)

Purists will point out that no big green group is clean, since virtually every one takes money from foundations built on fossil fuel empires – foundations that continue to invest their endowments in fossil fuels today. It's a fair point. Consider the largest foundation of them all: the Bill & Melinda Gates Foundation. As of December 2012, it had at least $958.6m – nearly a billion dollars – invested in just two oil giants: ExxonMobil and BP. The hypocrisy is staggering: a top priority of the Gates Foundation has been supporting malaria research, a disease intimately linked to climate. Mosquitoes and malaria parasites can both thrive in warmer weather, and they are getting more and more of it. Does it really make sense to fight malaria while fuelling one of the reasons it may be spreading more ferociously in some areas?

Clearly not. And it makes even less sense to raise money in the name of fighting climate change, only to invest that money in, say, ExxonMobil stocks. Yet that is precisely what some groups appear to be doing. Conservation International, notorious for its partnerships with oil companies and other bad actors (the CEO of Northrop Grumman is on its board, for God's sake), has close to $22m invested in publicly traded securities and, according to a spokesperson, "we do not have any explicit policy prohibiting investment in energy companies".

The same goes for Ocean Conservancy, which has $14.4m invested in publicly traded securities, including hundreds of thousands in "energy", "materials" and "utilities" holdings. A spokesperson confirmed in writing that the organisation does "not have an environmental or social screen investment policy". Neither organisation would divulge how much of its holdings were in fossil fuel companies or release a list of its investments. But according to Dan Apfel, executive director of the Responsible Endowments Coalition, unless an institution specifically directs its investment managers not to invest in fossil fuels, it will almost certainly hold some stock, simply because those stocks (including coal-burning utilities) make up about 13% of the US market, according to one standard index. "All investors are basically invested in fossil fuels," says Apfel. "You can't be an investor that is not invested in fossil fuels, unless you've actually worked very hard to ensure that you're not."

Another group that appears very far from divesting is the Wildlife Conservation Society. Its financial statement for fiscal year 2012 describes a subcategory of investments that includes "energy, mining, oil drilling, and agricultural businesses". How much of WCS's $377m endowment is being held in energy and drilling companies? It failed to provide that information despite repeated requests.

The WWF-US told me that it doesn't invest directly in corporations – but it refused to answer questions about whether it applies environmental screens to its very sizable mixed-asset funds. The National Wildlife Federation Endowment used to apply environmental screens for its $25.7m of investments in publicly traded securities, but now, according to a spokesperson, it tells its investment managers to "look for best-in-class companies who were implementing conservation, environmental and sustainable practices". In other words, not a fossil fuel divestment policy. Meanwhile, the Nature Conservancy – the richest of all the green groups – has at least $22.8m invested in the energy sector, according to its 2012 financial statements. Along with WCS, TNC completely refused to answer any of my questions or provide any further details about its holdings or policies.

It would be a little surprising if TNC didn't invest in fossil fuels, given its various other entanglements with the sector. A small sample: in 2010, the Washington Post reported that TNC "has accepted nearly $10m in cash and land contributions from BP and affiliated corporations"; it counts BP, Chevron, ExxonMobil and Shell among the members of its Business Council; Jim Rogers, CEO of Duke Energy, one of the largest US coal-burning utilities, sits on its board of directors; and it runs various conservation projects claiming to "offset" the carbon emissions of oil, gas and coal companies.

The divestment question is taking these groups off guard because for decades they were able to make these kinds of deals with polluters and barely raise an eyebrow. But now, it appears, people are fed up with being told that the best way to fight climate change is to change their light bulbs and buy carbon offsets while leaving the big polluters undisturbed. And they are raring to take the fight directly to the industry most responsible for the climate crisis.

Hannah Jones, one of the student divestment movement organisers, told me: "Just as our college and university boards are failing us by not actively confronting the forces responsible for climate change, so are the big corporate green groups. They have failed us by trying to preserve pristine pockets of the world while refusing to take on the powerful interests that are making the entire world unliveable for everyone." But, she added, "students now know what communities facing extraction have known for decades: that this is a fight about power and money, and everyone – even the big green groups – is going to have to decide whether they are with us, or with the forces wrecking the planet."

It doesn't seem like too much to ask. I mean, if the city of Seattle is divesting, shouldn't WWF do the same? Shouldn't environmental organisations be more concerned about the human and ecological risks posed by fossil fuel companies than they are by some imagined risks to their stock portfolios? Which raises another question: what are these groups doing hoarding so much money in the first place? If they believe their own scientists, this is the crucial decade to turn things around on climate. Is TNC planning to build a billion-dollar ark?

Some groups, thankfully, are rising to the challenge. A small but growing movement inside the funder world is pushing the big liberal foundations to get their investments in line with their stated missions – which means no more fossil fuels. It's time for foundations to "own what you own", says Ellen Dorsey, executive director of the Wallace Global Fund. According to Dorsey, her foundation, which has been a major funder of the coal divestment campaign, is now "99% fossil free and will be completely divested by 2014".

But convincing the biggest foundations to divest will be slow, and the green groups – which are at least theoretically accountable to their members – should surely lead the way. Some are starting to do just that. The Sierra Club, for instance, now has a clear policy against investing in, or taking money from, fossil fuel companies (it once didn't, which caused major controversy in the past). This is good news for the Sierra Club's $15m in investments in publicly traded securities. However, its affiliated organisation, the Sierra Club Foundation, has a much bigger portfolio – with $61.7m invested – and it is still in the process of drafting a full divestment policy, according to Sierra Club's executive director, Michael Brune. He stressed that "we are fully confident that we can get as good if not better returns from the emerging clean energy economy than we can from investing in the dirty fuels from the past".

For a long time, forming partnerships with polluters was how the green groups proved they were serious. But the young people demanding divestment – as well as the grassroots groups fighting fossil fuels wherever they are mined, drilled, fracked, burned, piped or shipped – have a different definition of seriousness. They are serious about winning. And the message to Big Green is clear: cut your ties with the fossils, or become one yourself.